AQA · Mathematics · 8300
GCSE Maths: Shortest distance from point to a line
Algebra · AQA GCSE Maths (8300). Cards are generated from the Corbettmaths textbook exercise for Shortest distance from point to a line (Corbettmaths Video 381), and every card links back to the exact page it came from. Practice tests are drawn from this deck’s own cards. Source material © Corbettmaths — used with attribution.
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What is the shortest distance from a point to a line measured along?
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The perpendicular from the point to the line — the segment meeting the line at a right angle.
How do you find the gradient of a line perpendicular to one with gradient m?
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Take the negative reciprocal: −1/m. The product of perpendicular gradients is −1.
What is the gradient of a line perpendicular to y = 2x − 3?
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−1/2 (the negative reciprocal of 2).
What is the distance formula between (x₁, y₁) and (x₂, y₂)?
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d = √((x₂ − x₁)² + (y₂ − y₁)²), from Pythagoras.
The point where the perpendicular from a point meets a line is called the ? of the perpendicular.
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foot
What is the shortest distance from (5, 0) to the line y = x + 1?
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3√2 ≈ 4.24. The perpendicular y = −x + 5 meets the line at (2, 3); distance = √(3² + 3²) = √18.
What is the shortest distance from (10, 11) to the line y = 6 − ½x?
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4√5 ≈ 8.94. The perpendicular y = 2x − 9 meets the line at (6, 3); distance = √(4² + 8²) = √80.
How do you find the shortest distance from a point to a line given as ax + by + c = 0?
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Rearrange to y = mx + c form (or use the formula |ax₀ + by₀ + c| / √(a² + b²)), then apply the perpendicular method.
What is the shortest distance from the origin to the line 2x + y + 5 = 0?
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√5 ≈ 2.24, since |2(0) + 0 + 5| / √(2² + 1²) = 5/√5 = √5.
How can the perpendicular distance from a point to a line be used to find a triangle's area?
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Take one side as the base (length via the distance formula) and the perpendicular distance from the opposite vertex to that side as the height, then use ½ × base × height.
If lines y = 2x − 4 and y = −3x + 11 meet at A, what is the shortest distance from A to y = x?
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√2 / 2 ≈ 0.71. A = (3, 2); the perpendicular y = −x + 5 meets y = x at (2.5, 2.5); distance = √(0.5² + 0.5²).
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